arXiv subjects

Amitay Kamber

Publications and source records attributed to Amitay Kamber.

11 recordsLinked to original sources

arXiv2024

Causal Responder Detection

We introduce the causal responders detection (CARD), a novel method for responder analysis that identifies treated subjects who significantly respond to a treatment. Leveraging recent advances in conformal prediction, CARD employs machine learning techniques to accurately identify responders while controlling the false discovery rate in finite sample sizes. Additionally, we incorporate a propensity score adjustment to mitigate bias arising from non-random treatment allocation, enhancing the robustness of our method in observational settings. Simulation studies demonstrate that CARD effectively detects responders with high power in diverse scenarios.

stat.ME↗

arXiv2023

Lifting all elements in $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$

We show that every element of $\mathrm{SL}_{n}(\mathbb{Z}/q\mathbb{Z})$ can be lifted to an element of $\mathrm{SL}_{n}(\mathbb{Z})$ of norm at most $Cq^2\log q$, while there exists an element such that every lift of it is of norm at least $q^{2+o(1)}$. This should be compared to the recent result that almost every element has a lift of norm bounded by $q^{1+1/n+o(1)}$. The main step in the proof is showing that for every $q$, there is a small element in $(\mathbb{Z}/q\mathbb{Z})^\times$ with a large $n$-th root, which is a result of independent interest.

math.NT↗

arXiv2022

Optimal Diophantine Exponents for $\mathrm{SL}(n)$

The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$, substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local $L^2$-norms of the Eisenstein series.

math.NT↗

arXiv2022

On the local $L^2$-Bound of the Eisenstein Series

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive groups. The method is based on Arthur's development of the spectral side of the trace formula, and ideas of Finis, Lapid, and M\"uller. As applications of our method, we prove the optimal lifting property for $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ for square-free $q$, as well as the Sarnak--Xue counting property for the principal congruence subgroup of $\mathrm{SL}_n(\mathbb{Z})$ of square-free level. This makes the recent results of Assing--Blomer unconditional.

math.NT↗

arXiv2022

Combinatorics via Closed Orbits: Number Theoretic Ramanujan Graphs are not Unique Neighbor Expanders

The question of finding expander graphs with strong vertex expansion properties such as unique neighbor expansion and lossless expansion is central to computer science. A barrier to constructing these is that strong notions of expansion could not be proven via the spectral expansion paradigm. A very symmetric and structured family of optimal spectral expanders (i.e., Ramanujan graphs) was constructed using number theory by Lubotzky, Phillips and Sarnak, and was subsequently generalized by others. We call such graphs Number Theoretic Ramanujan Graphs. These graphs are not only spectrally optimal, but also posses strong symmetries and rich structure. Thus, it has been widely conjectured that number theoretic Ramanujan graphs are lossless expanders, or at least unique neighbor expanders. In this work we disprove this conjecture, by showing that there are number theoretic Ramanujan graphs that are not even unique neighbor expanders. This is done by introducing a new combinatorial paradigm that we term the closed orbit method. The closed orbit method allows one to construct finite combinatorial objects with extermal substructures. This is done by observing that there exist infinite combinatorial structures with extermal substructures, coming from an action of a subgroup of the automorphism group of the structure. The crux of our idea is a systematic way to construct a finite quotient of the infinite structure containing a simple shadow of the infinite substructure, which maintains its extermal combinatorial property. Other applications of the method are to the edge expansion of number theoretic Ramanujan graphs and vertex expansion of Ramanujan complexes. Finally, in the field of graph quantum ergodicity we produce number theoretic Ramanujan graphs with an eigenfunction of small support that corresponds to the zero eigenvalue. This again contradicts common expectations.

math.CO↗

arXiv2022

Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

It was recently shown by Lubetzky and Peres (2016) and by Sardari (2018) that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in an optimal time and have an optimal almost-diameter. We show that this spectral condition can be replaced by a weaker condition, the Sarnak-Xue density property, to deduce similar results. This allows us to prove that some natural families of Schreier graphs of the $SL_2(\mathbb{F}_t)$-action on the projective line exhibit cutoff, thus proving a special case of a conjecture of Rivin and Sardari.

math.CO↗

arXiv2022

On Sarnak's Density Conjecture and its Applications

Sarnak's Density Conjecture is an explicit bound on the multiplicities of non-tempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue. The goal of this work is to discuss similar hypotheses, their interrelation and applications. We mainly focus on two properties - the spectral Spherical Density Hypothesis and the geometric Weak Injective Radius Property. Our results are strongest in the p-adic case, where we show that the two properties are equivalent, and both imply Sarnak's General Density Hypothesis. One possible application is that either the Spherical Density Hypothesis or the Weak Injective Radius Property imply Sarnak's Optimal Lifting Property. Conjecturally, all those properties should hold in great generality. We hope that this work will motivate their proofs in new cases.

math.NT↗

arXiv2021

Optimal Lifting for the Projective Action of $SL_3(Z)$

Let $ε>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $γ$ in $SL_3(Z)$, with coordinates bounded by $q^{1/3+ε}$, whose projection to $SL_{3}(F_q)$ sends $x$ to $y$. The exponent $1/3$ is optimal and the result is a high rank generalization of Sarnak's optimal strong approximation theorem for $SL_2(Z)$.

math.NT↗

arXiv2019

$L^p$ Expander Graphs

We discuss how graph expansion is related to the behavior of $L^{p}$-functions on the covering tree. We show that the non-trivial eigenvalues of the adjacency operator on aa $(q+1)$-regular graph are bounded by $q^{1/p}+q^{(p-1)/p}$ - the $L^{p}$-norm of the operator on the covering tree - if and only if properly averaged lifts of functions from the graph to the tree lie in $L^{p+ε}$ for every $ε>0$. We generalize the result to operators on edges and to bipartite graphs. The work is based on a combinatorial interpretation of representation-theoretic ideas.

math.CO↗

arXiv2017

Cutoff on Hyperbolic Surfaces

In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of $SL_{2}\left(\mathbb{Z}\right)$ and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis ([7]), who asked under what general phenomena cutoff exists.

math.GT↗

arXiv2016

Lp expander Complexes

We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of an Lp-expander complex. We calculate explicit spectral gaps on many combinatorical operators, on any Lp-expander complex. We associate with any complex a natural "zeta function", generalizing the Ihara-Hashimoto zeta function of a finite graph. We generalize a well known theorem of Hashimoto, showing that a complex is Ramanujan if and only if the zeta function satisfies the Riemann hypothesis.

math.CO↗